{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:WCFXNAS272A36HVT5QXLHLF4YL","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"08624dd3bc4815d6461887aec0ab305971b7276fcef63bf4a68f603fb1c2dd67","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-12-21T18:57:13Z","title_canon_sha256":"6a58e47043c38031197208882aa2bffdef335a3046720823fbd5597004990bdf"},"schema_version":"1.0","source":{"id":"1212.5555","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.5555","created_at":"2026-05-18T03:37:54Z"},{"alias_kind":"arxiv_version","alias_value":"1212.5555v1","created_at":"2026-05-18T03:37:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.5555","created_at":"2026-05-18T03:37:54Z"},{"alias_kind":"pith_short_12","alias_value":"WCFXNAS272A3","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"WCFXNAS272A36HVT","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"WCFXNAS2","created_at":"2026-05-18T12:27:25Z"}],"graph_snapshots":[{"event_id":"sha256:e7068fb0b829dcc8599c7126c8bd295944053db524c80ea94c1ffc619f9dda8f","target":"graph","created_at":"2026-05-18T03:37:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We consider the semilinear electromagnetic Schr\\\"{o}dinger equation (-i\\nabla+A(x))^{2}u + V(x)u = |u|^{2^{\\ast}-2}u, u\\in D_{A,0}^{1,2}(\\Omega,\\mathbb{C}), where $\\Omega=(\\mathbb{R}^{m}\\smallsetminus{0})\\times\\mathbb{R}^{N-m}$ with $2\\leq m\\leq N$, $N\\geq3$, $2^{\\ast}:= 2N/(N-2)$ is the critical Sobolev exponent, $V$ is a Hardy term and $A$ is a singular magnetic potential of a particular form which includes the Aharonov-Bohm potentials. Under some symmetry assumptions on $A$ we obtain multiplicity of solutions satisfying certain symmetry properties.","authors_text":"Andrzej Szulkin, M\\'onica Clapp","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-12-21T18:57:13Z","title":"Multiple solutions to nonlinear Schr\\\"odinger equations with singular electromagnetic potential"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.5555","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1a25f4b2f20ea07bd0c23744c0ca651cdb43330fb471241d0d3a88192a930611","target":"record","created_at":"2026-05-18T03:37:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"08624dd3bc4815d6461887aec0ab305971b7276fcef63bf4a68f603fb1c2dd67","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-12-21T18:57:13Z","title_canon_sha256":"6a58e47043c38031197208882aa2bffdef335a3046720823fbd5597004990bdf"},"schema_version":"1.0","source":{"id":"1212.5555","kind":"arxiv","version":1}},"canonical_sha256":"b08b76825afe81bf1eb3ec2eb3acbcc2cb244c17f32802cefc4b8c32fb978a04","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b08b76825afe81bf1eb3ec2eb3acbcc2cb244c17f32802cefc4b8c32fb978a04","first_computed_at":"2026-05-18T03:37:54.053122Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:37:54.053122Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Mcf/YsLQ1/KefJO1TJmCUjih1FFzx56FryaOfSITxzDXqbxUwS5gVoo1ZTon3VRzKjpe/HR6ghO0AKB+KZNEBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:37:54.053852Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.5555","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1a25f4b2f20ea07bd0c23744c0ca651cdb43330fb471241d0d3a88192a930611","sha256:e7068fb0b829dcc8599c7126c8bd295944053db524c80ea94c1ffc619f9dda8f"],"state_sha256":"7b21f82683f3fbda1d9162c3cf35b4f0a64224f7ddbec818acfec1d0b674f4c7"}